12-05: Trigonometric Equations
Trigonometric Functions
Periodic function: a function where the y values repeat over regular x value intervals
Key properties
y = sinx
y = cosx
y = tanx
Properties of Secondary Trig Functions
y = cscx
y = secx
y = cotx
Transformations of Sine and Cosine
Graphing Sine and Cosine Functions
Mapping
(x,y) → (1/k x - p, ay + q)
SINE | COSINE |
---|---|
(0,0) → ( _ , _ ) | (0,1) → ( _ , _ ) |
(π/2, 1) → ( _ , _ ) | (π/2, 0) → ( _ , _ ) |
Average (sine), Max (cosine), Average, Min
Properties (Graph with x and y scale)
y-scale
avg = q
max = q + |a|
min = q - |a|
x-scale
period = 2π / k
QI (quarter intervals) = Period/4
avg (Sin), max (Cos), avg, min, (and so on)
Solving Trigonometric Equations
Strategies
Rearrange (isolate for trig ratio)
Factor (exponents, same ratios repeats)
Identities (multiple ratios)
CAST rule must always be considered
Solving for Angles:
Approx. | Exact |
---|---|
Decimals | No decimals |
Calculator: sin⁻¹( ), cos⁻¹( ), tan⁻¹( ) | Special Triangles, main coordinates |
Solving Trigonometric Equations
Strategies
Rearrange - isolate for trig ratios
Factor - exponents, same ratio repeats
Identities - multiple ratios
In all cases: must consider CAST rule
Applications and IROC
AROC: average rate of change - slope of secant of curve
IROC: instantaneous rate of change - slope of tangent line (+ and - 0.001)
slope = ∆y/∆x
slope =( y2 - y1)/(x2 - x1)
12-05: Trigonometric Equations
Trigonometric Functions
Periodic function: a function where the y values repeat over regular x value intervals
Key properties
y = sinx
y = cosx
y = tanx
Properties of Secondary Trig Functions
y = cscx
y = secx
y = cotx
Transformations of Sine and Cosine
Graphing Sine and Cosine Functions
Mapping
(x,y) → (1/k x - p, ay + q)
SINE | COSINE |
---|---|
(0,0) → ( _ , _ ) | (0,1) → ( _ , _ ) |
(π/2, 1) → ( _ , _ ) | (π/2, 0) → ( _ , _ ) |
Average (sine), Max (cosine), Average, Min
Properties (Graph with x and y scale)
y-scale
avg = q
max = q + |a|
min = q - |a|
x-scale
period = 2π / k
QI (quarter intervals) = Period/4
avg (Sin), max (Cos), avg, min, (and so on)
Solving Trigonometric Equations
Strategies
Rearrange (isolate for trig ratio)
Factor (exponents, same ratios repeats)
Identities (multiple ratios)
CAST rule must always be considered
Solving for Angles:
Approx. | Exact |
---|---|
Decimals | No decimals |
Calculator: sin⁻¹( ), cos⁻¹( ), tan⁻¹( ) | Special Triangles, main coordinates |
Solving Trigonometric Equations
Strategies
Rearrange - isolate for trig ratios
Factor - exponents, same ratio repeats
Identities - multiple ratios
In all cases: must consider CAST rule
Applications and IROC
AROC: average rate of change - slope of secant of curve
IROC: instantaneous rate of change - slope of tangent line (+ and - 0.001)
slope = ∆y/∆x
slope =( y2 - y1)/(x2 - x1)